This book focuses on quantum groups, i.e., continuous deformations of Lie groups, and their applications in physics. These algebraic structures have been studied in the last decade by a growing number of mathematicians and physicists, and are found to underlie many physical systems of interest. They do provide, in fact, a sort of common algebraic ground for seemingly very different physical problems. As it has happened for supersymmetry, the q-group symmetries are bound to play a vital role in physics, even in fundamental theories like gauge theory or gravity. In fact q-symmetry can be considered itself as a generalization of supersymmetry, evident in the q-commutator formulation. The hope that field theories on q-groups are naturally reguralized begins to appear founded, and opens new perspectives for quantum gravity. The topics covered in this book include: conformal field theories and quantum groups, gauge theories of quantum groups, anyons, differential calculus on quantum groups and non-commutative geometry, poisson algebras, 2-dimensional statistical models, (2+1) quantum gravity, quantum groups and lattice physics, inhomogeneous q-groups, q-Poincaregroup and deformed gravity and gauging of W-algebras.

The CXXVII Course of the International School of Physics “Enrico Fermi” was held in Villa Monastero, Varenna, Italy, June 28-July 8, 1994.

The School focused on quantum groups, i.e. continuous deformations of Lie groups, and their applications in physics. These algebraic structures have been studied in the last decade by a growing number of mathematicians and physicists, and are found to underlie many physical systems of interest.

They do provide, in fact, a sort of common algebraic ground for seemingly very different physical problems, such as 2-dimensional and (2 + 1)-dimensional quantum gravity, 2-dimensional statistical models for condensed matter (spin chains, anyons, Hubbard model), integrable models, conformal field theories, Chern-Simons theories, discretized field theories, gauging of W-algebras, deformed gauge and gravity theories. This list is not complete, and keeps expanding.

Most of these subjects have been discussed in the School, as can be seen from the program of the lectures. The fruitful interplay between the formal mathematical approach and the (more “pragmatical”) physicists' point of view has engendered interesting and lively debates.

Villa Monastero, with its beautiful garden on the Como lake, has offered a perfect setting for discussions also outside the lecture hall. Most of the experts in the field were present as lecturers, including the almost complete "St. Petersburg School".

As it has happened for supersymmetry, the q-group symmetries are bound to play a vital role in physics, even in fundamental theories like gauge theory or gravity. In fact q-symmetry can be considered itself as a generalization of supersymmetry, evident in the q-commutator formulation. The hope that field theories on q-groups are naturally regularized begins to appear founded, and opens new perspectives for quantum gravity.

Our warmest thanks are due to all lecturers, for their effort to be pedagogical and their availability to students.

On behalf of all the participants, we wish to express our thanks to the SIF Secretary, Mrs. E. Mazzi, and her collaborators. The four vestals of Villa Monastero solved every logistic problem with ease and gentleness, besides organizing unforgettable parties.

Last but not least, we thank Mr. P. Papali and Mrs. C. Vasini of the Editorial Office for the careful editorial work on these Proceedings.